On Computing the Fuzzy Weighted Average Using the KM Algorithms

Transcription

1 O Computig the Fuzzy Weighted Average Usig the KM Algorithms Feilog iu ad Jerry M Medel Sigal ad Image Processig Istitute, Departmet of Electrical Egieerig Uiversity of Souther Califoria, 3740 McClitock Ave, os Ageles, CA feilog@uscedu/medel@sipiuscedu Abstract By coectig the fuzzy weighted average (FWA ad the geeralized cetroid (GC of a iterval type-2 fuzzy set we have arrived at a ew α- cut algorithm for solvig the FWA problem, oe that is mootoically ad super-expoetially coverget Our ew algorithm uses the Karik-Medel (KM algorithms to compute the FWA α-cut ed-poits No faster algorithm for solvig this problem exists to-date Keywords: Fuzzy weighted average, cetroid, KM algorithms, iterval type-2 fuzzy set 1 Itroductio The fuzzy weighted average (FWA, which is a weighted average ivolvig type-1 fuzzy sets, as explaied mathematically below, is useful as a aggregatio (fusio metho situatios where decisios ( x i ad expert weights ( are modeled as type-1 fuzzy sets, or where the decisios are modeled as either crisp umbers or iterval sets, ad the expert weights are still modeled as type-1 fuzzy sets Sometimes the same or similar problem is solve differet settigs This is a paper about such a situatio, amely computig the FWA, which is a problem that has bee studie multiple criteria decisio makig ([1]-[4], [7], [8], ad computig the geeralized cetroid of a iterval type-2 fuzzy set (herei referred to as the geeralized cetroid GC, which is a problem that has bee studie rule-based iterval type-2 fuzzy logic systems ([5], [9], [10] Here we demostrate how very sigificat computatioal advatages are obtaied for solvig the FWA problem by usig the GC algorithms developed by Karik ad Medel 1 To begi, cosider the followig weighted average: y =! x i! = f (w 1,,w, x 1,, x (1 I (1, are weights that act upo attributes x i Normalizatio is achieved by dividig the weighted umerator sum by the sum of all of the weights Without ormalizatio, too much emphasis would be placed o oe attribute over aother It is the ormalizatio that makes the calculatio of y very challegig i the FWA I (1 we are itereste the case whe some or!x i are type-1 (T1 fuzzy umbers, ie each x i is described by the membership fuctio (MF of a T1 fuzzy set (FS, µ Xi (x i, where this MF must be pre-specified, ad some or! are also T1 fuzzy umbers, ie each is described by the MF of a T1 FS, µ Wi (, where this MF must also be pre-specified y is a T1 FS, Y FWA with MF µ YFWA (y, but there is o kow closedform formula for computig µ YFWA (y! " cuts, a! " cut Decompositio Theorem [6] of a T1 1 These algorithms are ofte referred to i the T2 FS literature as the KM algorithms We shall also refer to them i this way

2 FS, ad a variety of algorithms ([1]-[4], [7], [8] ca be used to compute µ YFWA (y 2 Previous Approaches Begiig i 1987, various solutios to compute µ YFWA (y have bee proposed Dog ad Wog [1] were apparetly the first to develop a method for computig the FWA Although their algorithm is based o! " cuts ad a! " cut Decompositio Theorem [6], it is very computatioally iefficiet, because it uses a exhaustive-search iou ad Wag [8] did some importat aalyses that led to a improved algorithm (IFWA that drastically reduced the computatioal complexity of Dog ad Wog s algorithm They were the first to observe that (for each! " cut sice the x i appear oly i the umerator of (1, oly the smallest values of the x i are used to fid the smallest value of (1, ad oly the largest values of the x i are used to fid the largest value of (1 (Karik ad Medel [5], who were uaware oiou ad Wag s work, based their algorithms o the same observatios ee ad Park [7] preseted a efficiet FWA algorithm (EFWA that built o the observatios oiou ad Wag The total computatioal complexities of the three algorithms for m! " cuts are: FWA- O(m2 2, IFWA- O(m 2, ad EFWA- O(2m l These are ot the oly methods that have bee published Other methods are i [2]-[4] Except for [2], all of these methods are based o! " cuts ad a! " cut Decompositio Theorem Of the existig algorithms that use! " cuts, the EFWA represets the most computatioally efficiet oe to compute the FWA 3 The KM Algorithms for Computig the FWA As i the previous! " cut based methods, we begi by discretizig the complete rage of the membership [0, 1] of the fuzzy umbers ito m α-cuts,! 1,,! m, where the degree of accuracy depeds o the umber of α-cuts, ie m The, for each! j, we fid the correspodig itervals for X i i x i ad W i i, the ed-poits of which are [a i,b i ] ad [, ], respectively ( i = 1,, compute ( j = 1,, m where = f U = As i [7], [8], we Y FWA = " # f, f R % (2 mi " #[, ] max " #[, ] a i b i (3 (4 after which we defie the idicator fuctio + I! j (y = 1 "y # % f, f R & ' Y FWA (5 0 "y ( % f, f R &,+ ' from which we ca the costruct µ YFWA (y, as µ YFWA (y = sup " j I " j (y (6 Y!" j ( j=1,,m FWA What distiguishes our approach from earlier approaches are the ways i which we compute ad f R We have recogized that ad f R are precisely the same calculatios as eede the computatio of the GC of a iterval T2 FS [5], [9] i (3 ad f U i (4 ca be computed as (for otatioal coveiece we do ot show the explicit depedece of quatities o! j : = f U = mi! "[, ] max! "[, ] # a i # # b i # = = k # a i + a i k # + # i=k +1 # i=k +1 k U# b i + b i k U# + # i=k U +1 # i=k U +1 (7 (8 where switch poits k U ad k are computed usig the KM algorithms, which are stated ext Because of space limitatios, we caot provide derivatios (our joural versio of this paper will iclude them

3 31 KM Algorithm for k U ad f U 0 Rearragemet for the f U calculatios: I (1, x i are replaced by the b i, where b i are arrage ascedig order, ie b 1 < b 2 <! < b Associate the respective with its (possibly re-labeled b i I the remaiig steps we shall assume that the otatio use (8 is for the re-ordered (if ecessary ad b i 1 Iitialize for i = 1,2,,, ad the compute f U! = " b i " (9 Two ways for iitializig = ( + / 2 for are: (a i = 1,2,,, ad (b =, i! ( +1 / 2 "# % ad =, i >!" ( +1 / 2#, where!" # deotes the first iteger equal to or smaller tha 2 Fid k (1 k 1 such that b k! f U " < b k+1 3 Set = for i! k ad = for i! k +1, ad compute f U!!(b k " f U!!(k = k # b i + # b i k i=k+1 # + # i=k+1 (10 4 Check if f U!!(k = f U! If yes, the stop f U!!(k is the maximum value of! b i! which equals f U, ad k equals the switch poit k U If o, go to Step 5 5 Set f U! equal to f U!!(k ad go to Step 2 32 KM algorithm for k ad 0 Rearragemet for the calculatios: I (1, x i are replaced by the a i, where a i are arrage ascedig order, ie a 1 < a 2 <! < a Associate the respective with its (possibly re-labeled a i I the remaiig steps we shall assume that the otatio use (7 is for the re-ordered (if ecessary ad a i 1 Iitialize (i either of the two ways listed after (9 for i = 1,2,,, ad the compute! = " a i " (11 2 Fid k (1 k 1 such that a k! " < a k+1 3 Set = for i! k ad = for i! k +1, ad compute!!(a k "!!(k = k # a i + # a i k i=k+1 # + # i=k+1 (12 4 Check if!!(k =! If yes, the stop!!(k is the miimum value of a! i! w i ad equals f, ad k equals the switch poit k If o, go to Step 5 5 Set! equal to!!(k ad go to Step 2 4 Iterpretatio of the KM Algorithms We state three properties for!!(k so that the KM algorithm ca be provided with a graphical iterpretatio for its computatio Similar properties exist for f U!!(k It is assumed that a i have bee sorte icreasig order so that a 1! a 2!!! a Additioally, recall that the miimum of!!(k occurs at k = k ; hece, f =!!(k k is ccalled the switch poit for the miimum calculatio Property 1 (ocatio of the miimum: Whe k = k, the a k! ""(k < a k +1 where!! (k is computed by settig k = k i (12, ad mi ( f!! (k = f!! (k = f k=1,, This property locates the miimum by fidig the miimum of!!(k, k = 1,,, eg i Fig 1, the miimum of!!(k (k = 1,, is!!(4, so that k = 4 Observe that a 4! ""(4 < a 5, which locates the miimum

4 Property 2 (ocatio of!!(k i relatio to the lie y = a k :!!(k lies above the lie y = a k whe a k is to the left of, ad lies below the lie y = a k whe a k is to the right of, ie!!(k > a " #% k, whe a k < " &% < a k, whe a k > (13 This property provides iterestig relatios betwee a k ad!!(k o both sides of the miimum f, eg i Fig 1!!(k lies above a k to the left of f! " ##(4, at lies below a k to the right of f! " ##(4 This property does ot imply!!(k mootoic o either side of ; but, it does demostrate that!!(k caot be above the lie y = a k to the right of Property 3 is about the mootoicity of!!(k It will show, eg that!!(k is mootoically o-decreasig to the right of f ; but, this could occur i two very differet ways, amely,!!(k could be above the lie y = a k or below that lie to the right of Property 2 rules out the former Property 3 (Mootoicity of!!(k : It is true that!!(k "1 #!!(k whe a k < %!!(k +1 #!!(k whe a k > &' (14 This property shows that!!(k is a mootoic fuctio (but ot a strictly-mootoic fuctio o both sides of the miimum of the FWA For example, i Fig 1!!(k mootoically decreases to the left of f! " ##(4, whereas, it mootoically icreases to the right of f! " ##(4 Fig 1 provides a graphical iterpretatio of the KM algorithm that computes f The large dots are plots of!!(k for k = 1,,9 ; ote that k is associated with the subscript of a k The 45- degree lie y = a k is show because of the is computatios i Step 2 of the KM algorithm! has bee chose accordig to Method (b that is stated below (9 Because = 9,!" ( +1 / 2# = 5, which is why! is located at a 5 After! has bee computed, the a horizotal lie is draw util it itersects y = a k By virtue of Step 2 of the algorithm, we see that a 4! " < a 5, ad we the slide dow the 45- degree lie y = a k util we reach a 4, at which poit!!(4 is computed This is the vertical lie from a 4 that itersects a large dot Because!!(4 "!, the algorithm the goes through aother complete iteratio before it stops, at which time!!(4 has bee determied to be For this example, the KM algorithm coverges i two iteratios 5 Properties of the KM Algorithms Recetly, Medel ad iu [10] have prove that: 1 The KM algorithms are mootoically coverget 2 The KM algorithms are superexpoetially coverget et j deote a iteratio couter ( j = 1,2, A formula for j as a fuctio of prescribed bits of accuracy,!, is derive [10] Because we use this formula i Sectio 7 to support our simulatios, we state it ext for A comparable formula exists for f U et!(0 deote the first value of! as give i (11, ad, 2! " f ##(1 f #(0 f % 1 (15 Super-expoetial covergece 3 of the KM algorithm occurs to withi! bits of accuracy whe 2! " 1 follows from the mootoic covergece of the KM algorithms 3 Why this is super-expoetial covergece rather tha covergece is explaied fully i [10] It is evideced by the appearace of! 2 j i (17, the amplitude of whose logarithm is ot liear but is cocave upwards, which is idicative of a superexpoetial covergece factor

5 !!( j "!!( j "1 # (16 which, as prove i [10], is satisfied by the first iteger j! " 2 for which:!! 2 j "! 2( j"1 # %(0 " f (17 I geeral, this equatio has o closed form solutio for j; however, for small values of!, we ca drop the term! 2 j (which is much smaller tha the term! 2( j"1, ad solve for j! as: j! = 1+ 1 l 2 " l, 1 l# " l & # 0 - ( '(!(0 % f + 1 / 2 = 1+ 1, 1 " l 1+ l 2 l# " l & 0 - ( '(!(0 % f + 1 / 2 (18 j! = first iteger larger tha j " (19 We haste to poit out that the use of (18 ad (19 requires kowig the aswer as well as!!(1 The latter is oly available after the first iteratio of the KM algorithm; hece, the a priori use of these equatios is limited However, these equatios ca be used after-thefact (as we do i Sectio 7 to cofirm the very rapid covergece of the KM algorithms We have observed from may simulatios that for two sigificat figures of accuracy, the covergece of the KM algorithms occurs i 2-6 iteratios 6 EFWA Algorithms I Sectio 7 we compare covergece results for the KM algorithms with the EFWA algorithms Based o our derivatio of the KM algorithms, it is relatively easy to uderstad the followig: 61 EFWA Algorithm for f U 0 Rearragemet for the f U calculatios: Same as Step 0 i the KM Algorithm 1 Iitialize first = 1 ad last = 2 et k =!" 1 2 ( first + last # ad compute f U!!(k i (10 3 If b k! f U ""(k < b k+1, stop ad set f U = f U!!(k ad k = k U ; otherwise, go to Step 4 4 If f U!!(k " b k, set first = k + 1 ; otherwise, set last = k Go to Step 2 62 EFWA Algorithm for 0 Rearragemet for the calculatios: Same as Step 0 i the KM Algorithm 1 Iitialize first = 1 ad last = 2 et k =!" 1 2 ( first + last # ad compute!!(k i (12 3 If a k! f ""(k < a k+1, stop ad set f =!!(k ; otherwise, go to Step 4 4 If!!(k " a k, set first = k + 1 ; otherwise, set last = k Go to Step 2 Fig 2 provides a graphical iterpretatio of the EFWA algorithm that computes the same f as i Fig 1 As i Fig 1, the large dots are plots of!!(k for k = 1,,9 The 45-degree lie y = a k is show because of the computatios i Step 3 of the EFWA algorithm Sice = 9,!" ( +1 / 2# = 5, so that k = 5, which is why!!(5 has bee located at a 5 For this value of!!(5 we fail the test i Step 3 [ie,!!(5 is ot betwee a 5 ad a 6 ] ad as a result of Step 4 [ie,!!(5 < a 5 ] we set last = 5 Returig to Step 2, we fid k = 3 ad the compute!!(3, which is why there is a lie with a arrow o it from!!(5 to!!(a 3 =!!(3 For this value of!!(3 we agai fail the test i Step 3 [ie,!!(3 is ot betwee a 3 ad a 4 ] ad as a result of Step 4 [ie,!!(3 > a 3 ] we set first = 4 Returig to Step 2, we fid k = 4 ad the compute!!(4, which is why there is a arrow o it from!!(3 to!!(a 4 =!!(4 For this value of!!(4 we pass the test i Step 3 [ie,!!(4 is betwee a 4 ad a 5 ], stop ad set f =!!(4 This example, which is the same oe as i Fig 1, has take the EFWA three iteratios to coverge whereas it took the KM algorithm two iteratios to coverge, 7 Experimetal Results I this sectio we preset simulatio results i which we compare the covergece umbers for the KM ad EFWA algorithms Because, for each! " cut, both algorithms cosist of two

6 parts, oe for computig f U ad oe for computig, ad each part is essetially the same, we oly performed our simulatio study for computig There are may ways i which algorithm compariso experimets could be desiged, ad we do ot claim to have examied all of them I our simulatios we focused o radom desigs i which for a fixed value of (the umber of terms i the FWA ad! we chose each a i ( i = 1, 2,, ad the iterval ed-poits for each ( ad, i = 1, 2,, radomly accordig to a prescribed distributio We did this for = 2, 4, 6,, 200 ad for uiform, expoetial, Gaussia, Chi-squared, ad ocetral T-distributios The three radom sequeces a i, ad were geerated usig the same distributio but with differet seed umbers, ad for all the distributios a i![0,10],![0,1] ad![0,1] The a i were the sorted i icreasig order, ad the correspodig ad always satisfied the requiremets that! 0,! 0 ad! For each value of ad each distributio, 100 Mote-Carlo simulatios were performed ad the umbers of iteratios for both the KM ad EFWA to coverge to were recorded We the computed the mea ad stadard deviatio of the covergece umbers Because the results from these experimets looked very similar regardless of a specific distributio, we oly show our results here for the uiform distributio Parameters a i, ad were geerated by Matlab fuctio rad( usig differet seeds, respectively All a i values were scaled by 10, so that!a i "[0,10] Our results are summarize Fig 3 Note that covergece umber refers to the umber of iteratios it takes for a algorithm to STOP Note, also, that the curve labeled KM epsilo is for the KM algorithm that uses the approximate stoppig rule (16 for which we have chose! = 001 This stoppig rule is differet from the exact stoppig rule that is give i Step 4 of the KM algorithm Observe that: 1 The mea covergece umber for the KM algorithm is approximately two, at is "# (l!1% for the EFWA algorithm, where!" # deotes the first iteger equal to or larger tha The stadard deviatio of the covergece umber for the KM algorithm is approximately 05, at is approximately 12 for the EFWA algorithm 2 Accordig to cetral limit theory, the distributio of covergece umbers (for each is approximately ormal with mea ad stadard deviatio equal to the sample mea ad stadard deviatio sho Fig 2 We ca therefore use the sample mea + two times the sample stadard deviatio to evaluate the covergece umbers with 975% cofidece 4 Cosequetly we are 975% cofidet that the KM algorithm coverges i approximately three iteratios, ad that the EFWA algorithm coverges i approximately!" (l + 14# iteratios We see, therefore, that the KM algorithm is computatioally more efficiet tha the EFWA algorithm 3 Whe! 3, the KM algorithm eeds a smaller umber of iteratios tha does the EFWA algorithm Whe! 2, both algorithms eed the same umber of iteratios 4 The KM epsilo covergece umbers depicte Fig 3 agree with those obtaied from (18 ad (19, eg for = 200, we observed from ruig the KM algorithm that!(0 = 44077,!!(1 = , ad = 43783, so (! " %& ##(1 ' ( #(0 = ad l! = "61706 For! = 001, l{! /[ "(0 # f ]} = #10784, so that ( l 2 j = 1+ l 1+ l{! /[ "(0 # f ]} / l ( l 2 = 1+ l 1+ [#10784 #61706 = Because iteratios must be positive, this is a oesided cofidece iterval

7 Hece, j! = 2 which equals the true mea covergece umber of k = 2 8 Coclusios By coectig solutios from two differet problems the fuzzy weighted average (FWA ad the geeralized cetroid (GC of a iterval type-2 fuzzy set we have arrived at a ew α- cut algorithm for solvig the FWA problem, oe that coverges mootoically ad superexpoetially fast Our simulatios demostrate that for each α-cut, covergece occurs (to withi a 975% cofidece iterval i three iteratios No faster α-cut algorithm for solvig this problem exists to-date If 2m parallel processors are available, we ca use the ew KM α-cut algorithms to compute the FWA i three iteratios (to withi 975% cofidece, because the calculatios of f ad f U are totally idepedet, ad all m α-cut calculatios are also idepedet Research is uderway to exted the FWA from type-1 fuzzy sets to iterval type-2 fuzzy sets, somethig we call the fuzzy-fuzzy weighted average (FFWA [12] The FWA plays a major role i computig the FFWA We also believe that the FWA ca be used to perform typereductio for a geeral type-2 fuzzy set, somethig that we are also studyig Refereces [1] W M Dog ad F S Wog, Fuzzy weighted averages amplemetatio of the extesio priciple, Fuzzy Sets ad Systems, vol 21, pp , 1987 [2] D Dubois, H Fargier ad J Forti, A geeralized vertex method for computig with fuzzy itervals, Proc FUZZ IEEE 2004, Budapest, Hugary, pp , 2004 [3] Y-Y Guh, C-C Ho ad E S ee, Fuzzy weighted average: the liear programmig approach via Chares ad Cooper s rule, Fuzzy Sets ad Systems, vol 117, pp , 2001 [4] Y-Y Guh, C-C Ho, K-M Wag ad E S ee, Fuzzy weighted average: a maxmi paired elimiatio method, Computers Math Applic, vol 32, pp , 1996 [5] N N Karik ad J M Medel, Cetroid of a type-2 fuzzy set, Iformatio Scieces, vol 132, pp , 2001 [6] G J Klir ad B Yua, Fuzzy Sets ad Fuzzy ogic: Theory ad Applicatios, Pretice- Hall, Upper Saddle River, NJ, 1995 [7] D H ee ad D Park, A efficiet algorithm for fuzzy weighted average, Fuzzy Sets ad Systems, vol 87, pp 39-45, 1997 [8] T-S iou ad M-J J Wag, Fuzzy weighted average: a improved algorithm, Fuzzy Sets ad Systems, vol 49, pp , 1992 [9] J M Medel, Ucertai Rule-Based Fuzzy ogic Systems: Itroductio ad New Directios, Pretice-Hall, Upper Saddle River, NJ, 2001 [10] J M Medel ad F iu, Superexpoetial covergece of the Karik- Medel algorithms for computig the cetroid of a iterval type-2 fuzzy set, accepted for publicatio i IEEE Tras o Fuzzy Systems, 2005 [11] J M Medel, Fuzzy sets for words: a ew begiig, Proc of IEEE It l Cof o Fuzzy Systems, St ouis, MO, pp 37-42, May 2003 [12] D Wu ad J M Medel, The fuzzy-fuzzy weighted average (FFWA, to be submitted, 2005

8 (a Figure 1: A graphical iterpretatio of the KM algorithm that computes The solid lie show for y = a k oly has values at a 1,a 2,,a 9 (b Fig 3: (a Mea ad (b stadard deviatio of the covergece umbers whe parameters are uiformly distributed Fig 2: A graphical iterpretatio of the EFWA algorithm that computes The solid lie show for y = a k oly has values at a 1,a 2,,a 9